On the canonical formulation of electrodynamics and wave mechanics

The interaction of electromagnetic radiation with atoms or molecules is often understood when the timescale for the electromagnetic decay of an excited state is separated by orders of magnitude from the timescale of the excited state's dynamics. In these cases, the two dynamics may be treated separately and a perturbative Fermi golden rule analysis is appropriate. However, there do exist situations where the dynamics of the electromagnetic field and the atomic or molecular system occurs on the same timescale, e.g., photon-exciton dynamics in conjugated polymers and atom-photon dynamics in cold atom collisions.; Nonperturbative methods for the solution of the coupled nonlinear Maxwell-Schrodinger differential equations are developed in this dissertation which allow for the atomic or molecular and electromagnetic dynamics to occur on the same timescale. These equations have been derived within the Hamiltonian or canonical formalism. The canonical approach to dynamics, which begins with the Maxwell and Schrodinger Lagrangians together with a Lorenz gauge fixing term, yields a set of first order Hamilton equations which form a well-posed initial value problem. That is, their solution is uniquely determined and known in principle once the initial values for each of the associated dynamical variables are specified. The equations are also closed since the Schrodinger wavefunction is chosen to be the source for the electromagnetic field and the electromagnetic field reacts back upon the wavefunction.; In practice, the Maxwell-Schrodinger Lagrangian is represented in a basis of gaussian functions with different widths and centers. Application of the calculus of variations leads to a set of Euler-Lagrange equations that, for that choice of basis, form and represent the coupled first order Maxwell-Schrodinger equations. In the limit of a complete basis these equations are exact and for any finite choice of basis they provide an approximate system of dynamical equations that can be integrated in time and made systematically more accurate by enriching the basis. These equations are numerically implemented for a basis of arbitrary finite rank. The dynamics of the basis-represented Maxwell-Schrodinger system is investigated for the spinless hydrogen atom interacting with the electromagnetic field.